Root vs Leaf in Graph Theory

Understanding the fundamental differences between roots and leaves in tree structures

Core Concepts

Leaf (Terminal Node)

A leaf is a vertex with degree one. In simpler terms, it's a node that has exactly one connection to another node. Leaves are the endpoints of a tree structure.

Root

The root is the topmost node of a rooted tree, from which the rest of the tree descends. It is the only node that has no parent.

Key Differences

Feature	Root	Leaf
Degree	Can be any degree (1 or more)	Always has degree 1
Parent	Has no parent	Has exactly one parent
Children	Can have zero or more children	Has zero children
Hierarchy	The highest level (level 0)	The lowest level(s) of its branch
Requirement	Only exists in a rooted tree	Exists in both rooted and unrooted trees

Special Case: When Root and Leaf Are Equal

There is exactly one scenario where a root is also a leaf: in the trivial tree consisting of only one single vertex.

In this case, the single vertex is designated the root, and this single vertex also has a degree of zero (or one, depending on definition), making it a leaf.

As soon as a tree has two or more vertices, the root and leaf are distinct.

Tree Structure Visualization

In this diagram:

Root (A) - The topmost node with no parent

Leaves (D, E, F, G) - Nodes with degree 1, at the endpoints

Conclusion

Roots and leaves are not equal in graph theory. They represent opposite ends of the hierarchical structure in a rooted tree.

They are "bound" only in the sense that they are both essential, well-defined parts of the same tree structure, connected by the paths between them.